Monopoly, Oligopoly, and Strategic Pricing

Most real markets are not perfectly competitive. Monopoly (one seller), duopoly/oligopoly (a few major players), monopolistic competition (many differentiated sellers) are the main market structures, each with unique pricing strategies.

Monopoly: Analysis of Market Power

Monopolist's problem: max_q π = p(q)·q − c(q), where p(q) is the inverse demand function.

Necessary condition: MR = MC, where MR = ∂(p·q)/∂q = p + q·∂p/∂q = p(1 + 1/εₚ). Here, εₚ = ∂q/∂p · p/q < 0 is the price elasticity of demand.

Lerner index of market power:

L = (p − MC)/p = −1/εₚ ∈ [0, 1]

Breakdown: L = 0 → competitive price (p = MC). L → 1 → maximum monopoly power (with vertical demand). Optimal condition: price is set above MC proportionally to the inverse of elasticity.

Deadweight loss: at monopoly price pₘ > MC, production volume qₘ < q_c (competitive quantity). Harberger triangle: DWL = (pₘ − MC)(q_c − qₘ)/2. Estimates for the US: 0.1–2% of GDP (Harberger, Posner).

First-degree price discrimination (perfect): the monopolist charges each buyer their individual reservation price → extracts all consumer surplus. DWL = 0 (but CS = 0). Implementation: personalized pricing (Amazon, airline tickets with dynamic pricing).

Third-degree price discrimination: different prices for different segments. Optimum: MR₁ = MR₂ = MC → p₁/p₂ = (1 + 1/ε₂)/(1 + 1/ε₁). Higher price → less elastic segment (students receive discounts: |ε_st| > |ε_usual|).

Oligopoly: Cournot and Bertrand

Cournot model (quantity competition): n firms simultaneously choose quantities q₁,...,qₙ. Firm i's profit: πᵢ = p(Q)·qᵢ − cᵢ(qᵢ), Q = Σqⱼ. Response function: qᵢ = Bᵢ(Q₋ᵢ) = argmax πᵢ. Cournot NE: qᵢ = (a − c)/((n+1)b) for symmetric firms. Price: P = (a + n·c)/(n+1).

Bertrand model (price competition): firms simultaneously choose prices. Result: p* = MC (competitive price) even with two firms! Bertrand paradox. Resolutions: product differentiation, limited capacity (Edgeworth), dynamics.

Stackelberg equilibrium (leader-follower):

Leader (1) chooses q₁, anticipating the follower's response q₂ = B₂(q₁). Solution: q₁* = (a−c)/(2b), q₂* = (a−c)/(4b). Leader produces more (and earns more profit) — first-mover advantage.

Collusion and Its Instability

Cartel: firms agree to produce the total monopoly quantity Qₘ and split profit. Each firm gets πₘ/n. Incentive to deviate: with q_{-i} = Qₘ − qₘ/n, the optimal response of firm i yields profit π_dev > πₘ/n.

Repeated game: with infinite horizon and discount factor δ, grim trigger strategy sustains collusion if δ ≥ (π_dev − πₘ/n)/(π_dev − π_Nash). With δ = 0.9 and n = 2: threshold δ ≈ 0.5 → collusion is possible.

Markets with Product Differentiation

Hotelling model (1929): “linear city,” consumers are uniformly distributed on [0,1]. Two firms choose location and prices. Equilibrium: both locate in the center (minimum differentiation principle) — explains why all parties gravitate toward the center of the political spectrum.

Monopolistic competition (Chamberlin, 1933): many firms with differentiated goods. SR: p > MC (monopoly power). LR: π = 0 (entry of competitors reduces demand). LR equilibrium: p = AC > MC — small “brand” of differentiation.

Numerical Example

Cournot duopoly: P = 100 − 2Q, c₁ = c₂ = 10. Reaction functions: q₁ = (90 − 2q₂)/4, q₂ = (90 − 2q₁)/4. NE: q₁* = q₂* = 18, Q* = 36, P* = 100 − 72 = 28, π₁ = π₂ = (28−10)·18 = 324.

Monopoly outcome (cartel): Qₘ = 22.5, Pₘ = 55, πₘ/2 = (55−10)·22.5/2 = 506 > 324. Incentive to deviate: with q₂ = 11.25, best response q₁ = (90−22.5)/4 = 16.875. Profit upon deviation: P = 100 − 2(16.875+11.25) = 43.75, π_dev = (43.75−10)·16.875 ≈ 569.5 > 506/1.

Assignment: Market with P = 120 − Q. (1) Monopolist with c=20: find Pₘ, qₘ, πₘ, DWL. (2) Cournot duopoly (both with c=20): NE, profits, DWL. (3) Bertrand duopoly: equilibrium and profits. (4) Compare DWL in three cases. (5) For which n does Cournot yield DWL < 10% of monopoly DWL?